摺紙數列-相關問題探討

摘要或動機

1. 遊戲規則：將1～ 2^m × 2ⁿ的連續正整數，由上而下、由左而右依序填入 2^m × 2ⁿ的方格內。操作規則允許將2^m × 2ⁿ做往右或往左或往上或往下的完全對摺，直到操作至所有單位方格均疊成一行，此同時有數字也由上而下形成一數列。2. 本研究即是探討操作完成的數列之數量與數字間的關連性。3. 我們發現：(1) 數列之數量與巴斯卡三角形有關。(2) 形成的數列必符合內文的 [ R(L) 性質]、 [ D(U) 性質]、[ R&D 性質]、[D&R 性質]。
1. Rules of thegame: Fill in order the continuous positive integers 1~ 2^m × 2ⁿ, from top to bottom and from left to right in the 2^m × 2ⁿ check. The operational rule allows a complete fold of 2^m × 2ⁿ either rightward or leftward, or upward or downward, until all the check units pile up in a line. At the same time, all the integers form a series from top to bottom. 2. This study explores the relationship between the number of the series and the integers after the operation. 3. Our findings are: (1) The number of the series is related to Pascal triangles. (2) The series formed meet the properties mentioned in the study: [the property of R(L)], [the property of D(U)], [the property of R & D], and [the property of D & R].